Estimates of the approximation errors of the classes of continuous

In the metrics of continuous and non-decreasing functions φ(x), the following are obtained: a) estimates of the approximation of the classes of 1-periodic functions W^(2ν+1) Hω* (ν ∈ N), where ω(t) is an upwardly convex modulus of continuity, and by interpolating the functions f(t) ∈ W^(2ν+1) Hω*; b) by piecewise constant functions σ_n (f,t) in the integral metric L_p (0 < p < ∞); c) by piecewise constant functions σ_n (f,t) in a uniform metric. Estimates of the approximation errors of the classes of 1-periodic functions from the classes W^(2ν+1) Hω*  (ν ∈ N), where ω(t) is the convex upward modulus of continuity, by the piecewise constant functions σn(f, t) in the integral and uniform metrics Lp (0 < p < ∞). Estimates are expressed in terms of the function Ω_2v(w, t). The accuracy of the error estimates of the obtained approximations has been clarified. The theorem on the connection between the continuous and monotonically increasing function φ(x) ∈ Ф on the interval [0, ∞) and any function W^(2ν+1) Hω* (ν∈N)  and n = 2, 3, …, ∞ has been proved; as well as two lemmas and two consequences from the theorem. The results of the conducted research are a kind of extension of previously known results of approximation of functions to classes of 1-periodic functions    and more general spaces φ(L). It is proved that the obtained estimates are non-improvable for n = 2m (m ∈ N) on the entire class W^(2ν+1) Hω*. The new results of the function approximation theory obtained in the course of the study can be used for further practical applications, in particular, in the wavelet theory for the analysis of frequency components of signals (time-dependent functions) by methods similar to the Fourier transform. An applied aspect of the use of the obtained scientific results is also the possibility of applying estimates of approximation errors of the theory of numerical methods in the construction of numerical algorithms and signal processing in circuit engineering.